590 research outputs found
Accelerated finite difference schemes for stochastic partial differential equations in the whole space
We give sufficient conditions under which the convergence of finite
difference approximations in the space variable of the solution to the Cauchy
problem for linear stochastic PDEs of parabolic type can be accelerated to any
given order of convergence by Richardson's method.Comment: 24 page
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
Accelerated spatial approximations for time discretized stochastic partial differential equations
The present article investigates the convergence of a class of space-time
discretization schemes for the Cauchy problem for linear parabolic stochastic
partial differential equations (SPDEs) defined on the whole space. Sufficient
conditions are given for accelerating the convergence of the scheme with
respect to the spatial approximation to higher order accuracy by an application
of Richardson's method. This work extends the results of Gy\"ongy and Krylov
[SIAM J. Math. Anal., 42 (2010), pp. 2275--2296] to schemes that discretize in
time as well as space.Comment: 29 page
Spatial approximation of nondivergent type parabolic PDEs with unbounded coefficients related to finance
We study the spatial discretisation of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients. The equation free termand the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach and approximate in space the PDE problem’s generalised solution, with the use of finite-difference methods.Therate of convergence is estimated.info:eu-repo/semantics/publishedVersio
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